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Nonconforming finite element method for coupled Poisson–Nernst–Planck equations
Author(s) -
Shi Xiangyu,
Lu Linzhang
Publication year - 2021
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22764
Subject(s) - superconvergence , mathematics , finite element method , interpolation (computer graphics) , backward euler method , projection (relational algebra) , poisson's equation , norm (philosophy) , uniqueness , mathematical analysis , operator (biology) , poisson distribution , nernst equation , euler equations , algorithm , physics , classical mechanics , quantum mechanics , statistics , motion (physics) , repressor , law , chemistry , biochemistry , political science , transcription factor , thermodynamics , gene , electrode
A nonconforming finite element method (FEM) is developed and investigated for the coupled Poisson–Nernst–Planck (PNP) equations with low orderEQ 1 rotelement. Then, by use of the special properties of this element, that is, the interpolation operator is equivalent to its projection operator, and the consistency error estimate can reach order of O ( h 2 ) which is one order higher than that of its interpolation error estimates when the exact solution belongs to H 3 (Ω) , the superclose estimates of order O ( h 2 ) and O ( h 2 + τ ) in the broken H 1 ‐norm are derived with new techniques for the semidiscrete scheme and backward Euler fully discrete scheme, respectively. Further, through employing interpolation postprocessing approach, the corresponding global superconvergence results are obtained. Finally, some numerical results are provided to confirm the theoretical analysis. It seems that our results have never been found in the existing literature. Here h and τ denote the mesh size and time step, respectively.